"Show that for every infinite language L, there exists a sub-language L'
of L that is not Turing-recognizable (specifically, L' is undecidable)"

I'm not sure if I fully understood the proof…what do you mean by "We found a correspondence between L' and B (the set of all infinite
binary sequences), which is an uncountable set."
Is L' the set of all sub-languages of L or is it a sub-language of L?
and how can you deduce from this correspondence that the set of sub-languages of L is uncountable? Did you assume that it was an onto correspondence?

1. If the proof is confusing you, consider the following proof:
"Since L is infinite, it is at least countable, hence, P(L) (L's power set) is uncountable , we know the set of all TMs is countable, so we can deduce that there is a language in P(L) which is not countable" ($L' \in P(L) \leftrightarrow L' \subseteq L$)
2. There are only finite many number of TMs with n' states, out of these there are surely finite many number of TMs with n' states which halt on epsilon, just pick the one that takes the most steps.